I've noticed that the way his tentacle-things are portrayed tends to vary from depiction to depiction.
- Some are kind of like octopus tentacles with or without the suction cups.
- Some are more like branches.
- Some are more like spider legs.
- Sometimes we even get liquidy or ribbon-like ones.
However, if you look at some of the original images....
They seem more stiff and weighted down, almost droopy.
This actually makes more sense, since he was also seen using them to lift himself up and walk around, and thus they would need to be stiff in order to support his weight, like so.
The thinner, more flexible tentacles of fan art and modern ARGs likely would not be able to support his weight, and would bend under pressure.
This is important to understand when trying to determine exactly what his tentacles may be made of and how they function.
Skeletal Support
Now, at first a Hydrostatic skeleton seems like it would be likely for how Slender Man's tentacles work, but as explained here while the hydrostatic skeleton is strong enough to pry things open (and likely kill), heals quickly, and is lightweight, it lacks structure and "does not have surfaces for the attachment of muscles or limbs."
It also cannot support the weight of the creature using it. I mean, seriously, have you ever seen an octopus stand up on land? Plus, also explained in the link, is Slenderman did have a hydrostatic skeleton, he would have to stay in moist areas to keep it from drying out, and the forest is not exactly ideal.
The link also states one of the cons of a hydrostatic skeleton as, "Limited strength: Terrestrial animals with hydrostatic skeletons cannot increase their body size as they would collapse under their own body weight." making many of Slenderman's supposed abilities impossible (such as his ability to change size, or expand his tentacles from out of his back)
And exoskeleton would next seem ideal, but alas, also is unlikely as it would be far too heavy.
An endoskeleton is all that remains, and it still doesn't seem to fit, unless the bone structure is spine-like. This would mean his tentacles are snake like, rather than seeming like tentacles at all.
That being said, snakes can actually stand up pretty well on their own, but have to keep a large part of their body on the ground for balance. Slender Man would have multiple snake-like appendages at his disposal, and thus be able to balance himself. Unlike the snake, there likely would be no ribs and no internal organs in Slender Man's tentacles, which would mean the endoskeleton can afford to be thicker.
This would also explain the droopiness, much like our limbs usually are drooped downward when we are not using them.